Appendix: On the Reduction of Syllogisms

Appendix — On the Reduction of Syllogisms

To “reduce” a syllogism means to restate it in the shape of the First Figure. Hence only syllogisms of the Second, Third, and Fourth Figure are reducible. The First Figure has the following four moods:

AAA EAE All EIO

Examples:

1. (AAA)
All men are mortal
John is a man
Therefore, John is mortal.
2. (EAE)
No man is a spirit
John is a man
Therefore, John is not a spirit.
3- (AH)
All men are mortal
Some rational beings are men
Therefore some rational beings are mortal.
4. (EIO)
No man is a spirit
Some rational beings are men
Therefore, some rational beings are not spirits.

(note: It will be recalled from the Chapter on the classification of propositions that the singular proposition is equivalated to the universal proposition since the definition of the universal proposition is that it takes its subject in full Extension; and the singular proposition meets this requirement. This note is added, lest the student be surprised to find such a proposition as “John is a man” listed as an A-proposition.)

Now, the AAA mood of the First Figure is indicated by a proper name which contains all these vowels. It is called “Barbara.”

The EAE mood of the First Figure is called, for like
reason, “Celarent.”
The AII mood of the First Figure is called “Darii.”
The EIO mood of the First Figure is called “Ferio.”

In like manner all the moods of the other Figures (Second, Third, Fourth) are given proper names, and the whole is set forth in a Latin mnemonic stanza. Many of the consonants in the names have meanings which we shall presently explain. The mnemonic is:

Barbara, Celarent, Darii, Ferio, _sunt prioris_; (I
Figure)
Cesare, Camestres, Festino, Baroco, _secundce_; (II Fig
ure)
_Tertia_: Darapti, Disarms, Datisi, Felapton,
Bocardo, Feri son, _habet_; (III Figure). _Quarta insuper_

addit:

Bramantip, Camenes, Dimaris, Fesapo, Fresison.
(IV Figure.)

Now all the moods of the Second, Third, and Fourth Figures cannot be indiscriminately reduced to any desired mood of the First; but in each case the mood of the First Figure to which reduction is to be made, will be determined by the mood and figure of the syllogism to be reduced. Therefore, in taking up any syllogism with the purpose of reducing it, the first thing to determine is the figure and the mood of this syllogism as it stands.

To identify the figure of a syllogism look to the position of the middle term in the premisses. If middle term is

1. subject of major premiss, predicate of minor—
syllogism is in the First Figure;
2. predicate of both premisses—syllogism is in the
Second Figure;
3. subject of both premisses—syllogism is in the
Third Figure;
4. predicate of major premiss, subject of minor—
syllogism is in the Fourth Figure.

To identify the mood look for the valid moods in the mnemonic line which indicates the figure already identified. Find the word (proper name) which has in order the vowels which stand for the propositions of the syllogism to be reduced.

Suppose we have this syllogism to reduce:

All men are mortal beings (A-proposition)
Some men are wise beings (I-proposition)
Therefore some wise beings are mortal beings
(I-proposition)

First, we find that the middle term men is subject of both premisses. Thus we know that the syllogism is in the Third Figure.

Next, we take up the mnemonic line for the Third Figure:

_Tertia_: Darapti, Disarms, Datisi, Felapton,
Bocardo, Ferison,

and we look for the word which has the vowels (All) which stand for the propositions of the syllogism to be reduced. These vowels must tbe found in order, i. e., A-I-I. The word “Datisi” contains these vowels in order. Thus we discover that our syllogism is in the Third Figure in Datisi. Now, to reduce it:

1. The syllogism is in Datisi, a name with the initial D. This means that it can be reduced only to that mood of the First Figure which begins in D, that is, to Darii.

2. Next we look for the consonants in the name Datisi. For in the mnemonic names certain consonants have value:

s means that the premiss designated by the vowel
preceding is to be converted simply;
p means that the premiss designated by the vowel
preceding is to be converted _per accidens_, or acci
dentally;
m means that the major and minor premisses are to
change places;
c means that conversion is to be indirect.

Now, in Datisi we find the letter “s” following the vowel which indicates the minor premiss. This premiss then is to be converted simply. No other change is indicated. We make the reduction as fdllows:

All men are mortal beings
Some wise beings are men
Therefore, some wise beings are mortal beings.

Take another example:

(A) All men are mortal beings
(A) All mortal beings are bodily beings
(I) Therefore, some bodily beings are men.

To reduce:

1. The figure is the Fourth, for the middle term is the predicate of the major premiss and the subject of the minor.

2. The mood is Bramantip, for this is the word in the mnemonic line for the Fourth Figure, which contains in order the vowels which designate the propositions of the syllogism (AAI). We conclude that reduction must be made to Barbara since Bramantip begins with B.

3. We find the significant consonants “m” and “p” in Bramantip. The consonant “m” tells us to transpose the premisses, making them “change places.” “p” since it refers to conclusion and not to a premiss may be ignored.

4. Making the reduction we have:

(A) All mortal beings are bodily beings
(A) All men are mortal beings
(A) Therefore, all men are bodily beings.

The student is to note that syllogisms of the Second Figure in Baroco, and those in the Third Figure in Bocardo, cannot be reduced directly, since they contain O-propositions and, as we have learned, these are not directly convertible.

Coffey, P., _The Science of Logic_, Longmans Green & Co.,
Crumley, Thomas, C.8.C., _Logic, Deductive and Inductive_,
Macmillan, 1926.
Donat, Joseph, S.J., _Logica_, Felician Rauch, Innsbruck,
Frick, Charles, S.J., _Logica_, Herder, 1921.
Gredt, Joseph, O.S.B., _Elementa Philosophiae Aristotelico-_
_Thomisticae_, Herder, 1921.
Jevons, W. J., _Elementary Lessons in Logic_, Macmillan,
Joseph, H. W. B., _Introduction to Logic_, Oxford Uni
versity Press, 1916.
Joyce, George Hayward, S.J., _Principles of Logic_, Long
mans Green & Co., 1920.
Lortie, Stanislaus, _Elementa Philosophiae Christianae_,
L’Action Social, Quebec, 1921.

Pesch, Tilmann, S.J., Institutiones Logicales, Herder, 1914. Reinstadler, Sebastian, Elementa Philosophiae Scholasticae,

Herder, 1923.
Turner, William, _Lessons in Logic_, Catholic Education
Press, Washington, 1911.
(Numbers refer to pages) Abstraction, 4, 8. Comparison, 4, 9.
mathematical, n. Complex propositions, 90.
metaphysical, 11, Composition and division, 169
physical, 10. ff.
Accident, Comprehension (Connota
predicable, 33. tion), 16.

predicamental, 36 f. of terms, 44. Action, 38. Concept, 13. Adversative propositions, 84 f. Conclusion, Introd, viii, n8f. Analogy, 44 f. Conditional propositions, 88 f.

of attribution, 45 f. Conditional syllogism, 122,
of proportion, 45. 140 f.

argument from, 162. Conjunctive propositions, 89. Analysis, 4, 8. Conjunctive syllogism, 122, Appellation, 52 f. 142 f. Apprehension, see Simple ApConnective propositions, 88 f.

prehension. Connotation (Comprehen-Argumentation, 107, nyff. see sion), 16.

also Syllogism. Contradiction, 94. Attention, 4, 8. Contrariety, 95. Attribute, 32 f. Conversion, 101 ff.

Copula, 73. Begging the Question, 173 f. Copulative propositions, 84. Being, 15. Criteriology, Introd. x. Categoremata, 30. Deduction, 115 f. Categorematic words, 42. Definition, 54 if. Categorical propositions, 83 ff. Demonstration, 166. Categorical syllogism, 121 f. Denotation (Extension), 17. Categories, 36 f. Description, 56. Causal propositions, 86. Dialectics, Introd. vii. Common term, 48. Dictum, 81. Comparative propositions, 87. Dictum de Nullo, n6.

Dictum de Omni, 115. Grades, Metaphysical, 16 f. Difference, see Specific DifferGrades of ideas, 10.

ence.
Dilemma, 158 f. Habit, 38.
Direct inference, _Introd, vii_, Horned Syllogism, The, 158.
93, 108. Hypothesis, 163 ff.

Direct intention, 14. Hypothetical propositions, 88. Direct Universal, 36 ff. Hypothetical syllogism, 122, Disjunctive propositions, 89 f. 140 ff.

Disjunctive syllogism, 122,
M3 Idea, i-68, _Introd, xv_,
Dividing Members, 61 ff. abstract, 23 f.
Division, 60. associable, 25.
clear, 22.
Enthymeme, 153 f. collective, 24 f.
Epichirema, 15Z. complete, 23.
Epistemology, _Introd, x,_ compound, 23.
Equipollence, 99 ff. concrete, 23.
Equivocal Terms, 44. confused, 23.
Equivocation, 167. constituents of, 17.
Error, 166. contradictory, 25 f.
Essence, 3, 19 f. contrary, 25.
Evading the Point, 171 ff. definition of, 19.
Example, 162. derivative, 22.
Exceptive propositions, 86 f. description of, 1 ff.
Exclusive propositions, 86. different, 25.
Extension (Denotation), 17. distinct, 22 f.
of terms, 44. formation of, 4-ff.
grades, 10.
Fact, scientific, 163. identical, 25.
Fallacy, 166 ff. incomplete, 23.
False Cause, 174 f. intuitive, 22.
Fancy, 19. mathematical, 11.
Figures of Syllogism, 146 ff. metaphysical, 11 f.
Formal Logic, _Introd, x._ negative, 26.
First intention, 14. obscure, 22.
Formation of Ideas, 4ff. opposed, 25.
particular, 24.
Genus, 31 f. physical, 10.
intermediate, 35. privative, 26.
proximate, 35. relative, 26.
remote, 35. simple, 23.
Idea (_continued)_ Mental Image, 13.
singular, 24. Mental Term, 14.
transcendental, 24. Metaphysical grades, 16 f.
universal, 24, 27 ff. Metaphysical parts, 16, 60.

vague, 23. Methods of Reasoning, 113 if. Ideation, see Idea, formation Middle Term, 118.

of. Minor Premiss, 119. Inference, _Introd_. vr'r. Minor Term, 118.
immediate or direct, 93, Modal Propositions, 81 if.
109. Mode, 82.
mediate or indirect, _see_ Modes of Predication, 30 if.
Reasoning. Moods of the Syllogism, Inferiors of Universal, 29. 149 if.
Individuating marks and notes, Multiple propositions, 90 f.
Induction, 113 if. Natural Logic, _Introd. xiii_. Intention, 14, 19. Notes,

Intentional Image, 19. accidental, 3. Intermediate Genus, 35. essential, 14 f. Intermediate Species, 35. individuating, 3. Inverse Ratio of ComprehenNotion, 13.

sion and Extension, 18.
Object of science, _Introd, xi_, Judgment, _Introd, xv._, 67-108. _xiv_.
definition of, 70 f. Opposition, 93 f.
description of, 69 f. Origin of Ideas, _see_ Idea, for

elements of, 71. mation of. Law, scientific, 163. Parable, 163. Laws of Thought, Introd. viii. Passion, 39.

Logic, Percept, 13.
Formal, _Introd, x._ Phantasm, 19.
Material, _Introd, x._ Physical parts, 16, 60.
Natural, _Introd, xiii_. Place, 38.

Logical Division, 59 if., 61 if. Poly syllogism, 156 f. Logical Square, The, 97. Porphyrian Tree, 34.

Posture, 38.

Major Premiss, 119. Predicables, The, 30. Major Term, 118. Predicamentals, The, 36 f. Material Logic, Introd. x. Predicate, 73. Mediate Inference, see ReaPremisses, Introd, viii, Ii8f.

soning. Principle of Division, 60 if. Proper term, 48. Quality, 37.
Property, 32 f. Quantity of propositions, 75. Proposition,
Real Division, 60.
adversative, 84 f.
Reasoning _Introd. vii-xv._, affirmative, 76.
107-180.
categorical, 83 f.
deductive, 115 ff.
causal, 86.
definition of, 113.
comparative, 87.
description of, 109 ff.
complex, 90.
elements of, 113.
compound, 85 ff.
inductive, 113 ff.
conditional, 88.
methods, 113 ff.
conjunctive, 89.
Reduction of syllogisms, 152. connective, 88.
_see also_ Appendix.
conversion of, 101 ff.
Reduplicative propositions, copulative, 84.
definition of, 87 f.
Reflection, 4, 8.
disjunctive, 89 f.
Reflex intention, 14.
elements of, 73.
Reflex Universal, 30 ff.
equipollence of, 99 ff.
Relation, 38.
exceptive, 86 f.
Relative propositions, 85 f. exclusive, 86.
Remote Genus, 35.
hypothetical, 88 f.
indefinite, 76 f. Science,

logical form of, 74. definition of, Introd. vii. modal, 81 ff. formal object of, Introd. multiple, 90 f. xi., f.

negative, 76. material object of, _Introd_. opposition of, 93 ff. _xl., f_.
particular, 76.
practical, _Introd. vii_.
properties of, 92 ff.
speculative, _Introd, vii_.
quality of, 75. Scientific Law, 163.
quantity of, 75. Second intention, 14.
reduplicative, 87 f. Sensation, 3 f.
relative, 85. Sign, 41 f.
simple, 76. arbitrary, 42.
singular, 76 f. conventional,
universal, 76 f. formal, 42.
Proximate instrumental, 42.
Genus, 35.
natural, 42.
Quadrilemma, 161. Similitude, 162.
Quality, 37. Simple Apprehension, _Introd_. Quality of propositions, 75. _XV._, 12 f.

Sophistry, 166. Term (continued) Sorites, 157. appellation of, 52 f. Species, collective, 48. intelligible, 12. common, 48. predicable, 31. complex, 47. Species expressa, 13 f. comprehension of, 44. Species impressa, 13. concrete, 47. Specific Difference, 32. definition of, 41. Square of Opposition (The equivocal, 44.

Logical Square), 97. extension of, 44. Subalternity, 96. gesticular, 43. Subcontrariety, 95. indefinite, 48. Subject of proposition, 73. major, 118 f. Subject of Universal, 29. mental, 14. Subordination of Genera and middle, u8f.

Species, 34 f. minor, u8f. Substance, 36. negative, 46. Supposition, 49 ff. oral, 42. Syllogism, 117 ff. particular, 48. categorical, 121. positive, 46. conditional, 122, 144 f. privative, 46. conjunctive, 122, 142 f. proper, 48. correctness of, 119, 121. simple, 47. definition of, 117 f. singular, 48. disjunctive, 122, 143 f. supposition of, 49 ff. elements of, n8f. universal, 44, 48. figures of, 146 ff. use of, 49 ff. hypothetical, 122. written, 43. imperfect, 153 ff. Theory, 163. laws of, 124 ff., 140 ff., 148. Thinking, Introd, viii’ee moods of, 149 ff. also Reasoning. propositions of, 118 f. Time, 38.

reduction of, 152, _see also_ Totality Divided, 61.
Appendix. Tree of Porphyry, 34. terms of, n8f. Trilemma, 161. truth of, 119, 121.
Syncategorematic words, 43. Universals, 27 ff. Synthesis, 4, 8. direct, 36 ff.
reflex, 30 ff. Term, 41-53.
abstract, 47. _Verbum Mentis_, 14. analogous, 44 f. Vital Representation,